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March 11, 2006

Remarks on String(n)

Posted by Urs Schreiber

Let PSpin(n) be the 2-group (→) whose nerve is the group String(n) (→).

I would like to understand if Stolz/Teichner’s conception (→) of String(n)-connections can be understood in terms of 2-connections in 2-bundles (→) which are associated by way of a 2-representation (→) to a principal PSpin(n)-2-bundle (→).

Here are some remarks.

Let me fix some notation.

For K a group, let Σ(K) (the “suspension of K”) be the category with a single object • and K=Hom(•,•).

Similarly, for G2 a 2-group, let Σ(G2 ) be the 2-category with a single object • and G2 =Hom(•,•).

For any category C, let Aut(C) be the 2-group whose objects are automorphisms of C and whose morphisms are natural isomorphisms of these automorphisms. In other words

(1)Σ(Aut(C))=AutCat(C).

A representation of a group K is a functor on Σ(K).

A representation of a 2-group G2 is a 2-functor on Σ(G2 ).

The main point of my remarks here is the simple but useful observation that every representation of a group K induces a representation of the 2-group Aut(K).

I’ll spell that out in detail in a moment. The relevance for the case at hand is the following.

The 2-group PSpin(n) sits inside Aut(Ω̂Spin(n)), where Ω̂Spin(n) is a Kac-Moody central extension of the loop group of Spin(n).

(2)PSpin(n)→Aut(Ω̂Spin(n)).

Therefore, the above statement implies that every representation of Ω̂Spin(n) induces a 2-representation of PSpin(n).

But Ω̂Spin(n) is represented in terms of unitary elements in von Neumann algebras (“positive energy reps of loop groups”). The induced 2-representation of PSpin(n) is, implicitly, pretty much what Stolz/Teichner are considering.

Here is what I mean by the 2-representation of Aut(K) induced by a representation of K. It is a simple variation on the theme of the construction of Aut(K) itself.

So let

(3)ρ:Σ(K)→AutVect(ℂn)⊂Vect

be a representation of H. Alternatively, think of a representation

(4)ρ:Σ(K)→AutHilb(H)⊂Hilb

on Hilbert spaces.

Denote by Im(ρ) the image category of ρ. Then we get a 2-representation

(5)ρ2 :Σ(Aut(K))→Aut(Im(ρ))⊂AutCat(Hilb)

as follows.

i) the image of the single object • is the image of ρ

(6)ρ2 (•)=Im(ρ)={H→ρ(k)H∣k∈K}.

(Here H denotes the vector space/Hilbert space on which K is represented by ρ.)

ii) the image of an automorphism •→g• of K is the functor g:Im(ρ)→Im(ρ)
acting as

(7)Hρ(k)↓H↦Hρ(g(k))↓H

iii) the image of a 2-morphism

(8)•→g•k′⇓•→g′•

is the natural ismorphism given by these naturality squares

(9)H→ρ(k′)Hρ(g(k))↓↓ρ(g′(k))H→ρ(k′)H.

(Notice that this is nothing but the image under ρ of the corresponding naturality squares in Aut(K)).

While this makes manifest sense only if ρ is faithful, one can check that it is also well defined in the general case.

Consider the example which is of interest here. PSpin(n) can be regarded as a sub-2-group of Aut(Ω̂Spin(n)).

Recall how Antony Wassermann explains that we get representations of these centrally extended loop groups (section 5.4 of Stolz/Teichner).

Let ρ be a projective unitary representation of ΩSpin(n) on some Hilbert space H

(10)ρ:ΩG→PU(H).

The projective unitary group is PU(H)=U(H)/T (T is the circle group), hence fits into the short exact sequence

(11)1 →T→U(H)→PU(H)→1 .

Let’s draw ρ into this diagram:

(12)1 →T→U(H)→PU(H)→1 ↑ρΩG

and consider the pullback of ρ to the left

(13)1 →T→U(H)→PU(H)→1 Id↑ρ̂↑↑ρT→Ω̂G→ΩG

thus obtaining a unitary representation

(14)ρ̂:Ω̂G→U(H)

of the Kac-Moody loop group on H.

Now think of the loops in ΩG as maps from the circle 𝕊1 into G
Then we get a vonNeumann algebra Aρ from this representation by looking at the closure of the image under ρ̂ of all loops that take values different from 1 only on the upper half circle I⊂𝕊1

(15)Aρ:=ρ̂(Ω̂IG)″.

(The two primes indicate taking the double commutant, which amouts to closing the image of ρ̂ in weak operator norm.)

So Aρ plays the role of the representation of Ω̂Spin(n).

The objects in PSpin(n) are based paths in Spin(n). Hence, in order to be able to follow the above prescription for the construction of representations of a 2-group, we need to check that the technical subtleties involved in the construction of Aρ still allow us to act with based paths in Spin(n) on Aρ. Indeed, this is the case (Stolz/Teichner, p. 83).

The trick is to regard an open path in Spin(n) as a map γ:I→Spin(n), where, recall, I was the upper half circle. Any such map may be extended to a map γ̂:𝕊1 →Spin(n) on all of 𝕊1 and then sent it with ρ̂ to U(H). By definition of Aρ, conjugating any element with such a lift

(16)Aρ∋a↦ρ̂(γ̂)aρ̂(γ̂−1 )

is independent of the choice of lift γ̂ of γ and hence well defined.

(What Stolz/Teichner discuss is that this construction even works on PU(Aρ).)

Hence, unless I am overlooking something, for every projective unitary representation of the loop group of Spin(n) we get a 2-representation of the 2-group PSpin(n) on
Aut(Σ(Aρ)), i.e. on the 2-category whose 1-morphisms are automorphisms of the category with single object H and vonNeumann operators Aρ as morphisms, and whose 2-morphisms are natural isomorphisms between these.

More precisely, let γ and γ′ be based paths in Spin(n) and let ℓ̂ be a centrally extended based loop in Spin(n) such that γ′=ℓ⋅γ. Then

(17)γ↓ℓ̂γ′

is a morphism in PSpin(n) and it is sent by our 2-representation induced by ρ to the natural isomorphism given for each morphism

(18)Ha↓H

by the naturality square

(19)H→ρ̂(ℓ̂)Hρ̂(γ̂)aρ̂(γ̂−1 )↓↓ρ̂(γ̂′)aρ̂(γ̂′−1 )Hρ̂(ℓ̂)→H.

This should mean that there should be a 2-connection on a String(n)-bundle obtained as follows.

From a principal PSpin(n)-2-bundle we obtain a String(n)-bundle using Branislav Jurčo’s construction (→). Following Stolz/Teichner, who observed that (p. 85)

(20)PU(H)→Inn(Aρ)↓↓String(n)→Aut(Aρ)↓↓Spin(n)→Out(Aρ)

we have an action of String(n) on Aρ and hence can form an associated vonNeumann algebra bundle. A 2-connection taking values in the above representation of the 2-group PSpin(n) would give the local surface transport in a locally trivialized version of this.

I won’t go into the details right now. I’ll just note that the “fake flatness” condition (→) on that 2-connection ensures that there is precisely a circle worth of surface transport between given fixed boundaries. This is precisely as in Stolz/Teichner’s formulation (paragraph below digram on p. 70).

Posted at March 11, 2006 1:28 PM UTC

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Read the post Remarks on 2-RepsWeblog: The String Coffee TableExcerpt: More on 2-representations of the String(n) group.Tracked: March 16, 2006 3:52 PM

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